Derivation of E=mc²

Albert Eisenstein is father of modern physics, his theories and postulates are still very difficult to understand after half century of his pass away. One formula which brings most horrible disaster in the world in the face of Hiroshima and Nagasaki, and  now the formula is giving power to quarter th of the world, The formula which changes the world of physics with two different things i.e. mass and energy to be one and same thing. This was starting of unification theory, latter on scientist thought that if electrostatics and magnetism are one and the same thing and mass and energy are one and the same, then why not all the other physical particle and forces are same. In this manner they started search for the Unification theory which now termed as The M-Theory.

That elegant formula is E=mc². Lets derive this formula using the modern physics approach.

We all know that De-Broglie was the first man who say that matter have dual nature. Some time it behaves like particle and sometime as wave, and light also has dual nature. Now one thing come into mind that if matter and light both have dual nature then they must be one and the same thing. This was the thought of Eisenstein.

Come to the topic, De-Broglie explained that if any thing that have momentum must have wave nature and wavelength is given by:-

here λ = Wave length of wave packet

h = plank's constant

p = momentum of wave packet

A wave packet is a terminology for representing a thing which is wave as well as particle. From the above equation it is clear that anything which has momentum must have associated wavelength.

So the reverse must be true, if anything has wavelength must have momentum too. And momentum means both velocity and mass. When it is concern about normal matter all the matter have momentum due to it's mass and velocity and associated wavelength too.

But one particle in nature have only wavelength but no mass, this is photon. A photon has zero rest mass but it have momentum, as it exert impulsive force to the objects it is falling on. So from the above equation photon must have momentum, which is true. But what about mass of photon ?

Since photon have wavelength, and it is move with the velocity of light thus it have frequency. And according to Plank's Quantum Theory Energy of wave packet has it's energy related to it's frequency as blow:-

 here E = Energy of wave packet

h = Plank's Constant

v= Frequency of wave packet

The above equation can be written in terms of wavelength, as 

From both the Quantum theory and De-Broglie equation we can easily say that energy of photon is directly proportional to the momentum of photon as:-

From the classical physics we know that momentum is product of mass and velocity. So if photon has momentum and energy which are related as follows then it must have equivalent mass which can produce such momentum for producing the energy given by above equation.

So let the equivalent mass of photon is m, which produce momentum of p, and because photon always run with the speed of light so p = mc. By placing value of p into above equation we get:-

From the formula it is clear that if any photon has energy E then it must have equivalent mass of m. The converse if also true that if any thing has mass of m it must have equivalent energy equal to E.

This is MASS ENERGY RELATION of EISENSTEIN.